Definition:Covolume of Lattice

Definition

Let $L$ be a lattice in $\R^n$.

Let $\left({v_1, \ldots, v_n}\right)$ be an ordered basis for $L$.

Let $v_i = \left({v_{i 1}, \ldots , v_{i n} }\right)$ for $i \in \left\{ {1, \ldots, n}\right\}$.

The covolume of $L$ is the determinant of the matrix:

$\begin{bmatrix}

v_{1 1} & v_{1 2} & \cdots & v_{1 n} \\ v_{2 1} & v_{2 2} & \cdots & v_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ v_{n 1} & v_{n 2} & \cdots & v_{n n} \\ \end{bmatrix}$

Also see

• Covolume of Lattice is Well-Defined

Sources

• 1999: J.H. Conway and N.J.A. Sloane: Sphere Packings, Lattices, and Groups (3rd ed.): Chapter $1$: $\S 1.2$